Problem 87 - Precalculus Student Solutions Manual 5th

Question

Write each statement as an absolute value equation or inequality. \(r\) is no less than 1 unit from 29.

Step-by-Step Solution

Answer

\(|r - 29| \geq 1\)

Step 1: Understanding the Problem

Identify what is being asked. The problem states that the value of \(r\) is no less than 1 unit from 29. This implies that the distance between \(r\) and 29 is at least 1 unit.

Step 2: Formulating the Absolute Value Expression

Recall that absolute value represents the distance from a point on a number line. The expression for the distance between \(r\) and 29 is given by \(|r - 29|\).

Step 3: Setting Up the Inequality

The problem states that the distance is no less than 1 unit. This translates to the inequality \(|r - 29| \geq 1\).

Key Concepts

distance on a number line

Understanding distance on a number line is essential for grasping absolute value inequalities. Imagine a number line, which is a straight line with numbers placed at equal intervals along its length. Each point on the line represents a different value. The **distance** between any two points on this number line is simply the absolute difference between their values.
For example, if you have numbers 5 and 8, the distance between them is \(|8 - 5| = 3|\).
This distance is always positive because distance cannot be negative. It tells you how far apart the numbers are.
In the context of our exercise, we are looking at the distance between a variable, \(r\), and a fixed number, 29. The distance needs to be at least 1 unit.

absolute value

The concept of **absolute value** is closely tied to distance. The absolute value of a number is the distance of that number from zero on the number line, without considering direction. For instance, the absolute value of both 7 and -7 is 7, because both are 7 units away from zero.
Mathematically, absolute value is represented with two vertical bars, like this: \(|x|\). Here, \(|x|\), where x can be any real number, denotes the absolute value of x.
In our exercise, the absolute value expression \(|r - 29|\) gives us the distance between r and 29. It doesn't matter if r is greater than or less than 29; it only measures how far away r is from 29.

inequalities

Inequalities are mathematical expressions that show the relationship between two values, where they are not necessarily equal. Inequalities use signs like \(<, >, \leq, \geq \).
• \(<\) means 'less than'
• \(>\) means 'greater than'
• \( \leq \) means 'less than or equal to'
• \( \geq \) means 'greater than or equal to'

In the given exercise, we need an inequality because we want to express that the distance between r and 29 is at least 1 unit.
The appropriate inequality for this situation is \(|r - 29| \geq 1|\). It says that the absolute value of the distance between r and 29 should be no less than 1.
This kind of inequality involving absolute value helps us capture a range of possible values for r.